Long Shank Ballcock

Hope you can help. I have an old toilet that I would guess is about 65 years old. The only id I can find on it says Integera. It is a single mold where the bowl and tank are one. I need to replace the ballcock but am unable to find a suitable replacement. The existing ballcock says Sherwood Detroit. The unique thing about it, is that the shank on the ballcock is 3 1/2 inches long. The longest shank I could find anywhere is a 2 3/4 inch. I have heard that at one time they made a shank extension, but again, no luck being able to track one down. I talked to many a local plumber and plumbing supply store here in Iowa and everyone says I'll have to replace the whole toilet. The toilet is original with the house (1940) and I would like to keep it because it works great and goes great with the era of the home.
Any ideas or suggestions where I can find a long shank ballcock or a shank extension? Thanks in adavance for your help.

The porcelain on the tank is very thick. When I first tried a standard stock ballcock, the shank was not even visible on the bottom of the tank. With a 2 3/4 inch shank, there is barely enough thread to even have the nut fit tight to fasten the ballcock to the bowl.
I would guess the procelain is 2 inches thick.